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Re: Mlodinow: We Invent the Laws of Nature

Posted: Tue Sep 19, 2023 10:10 pm
by Magnus Anderson
Iwannaplato wrote:We could identify two completely different things and say they are one thing. I am not sure this makes them one thing.
Suppose we have two different things A and B. We can come up with a symbol X and assign it the concept captured by the sentence "Either A or B". That would make both A and B an X. That would make them the same in this particular regard -- they would both be X. In other words, they would both be "Either A or B". And that has nothing to do with what anyone believes and what anyone desires. You're not free to decide that A isn't X or that B isn't X.

If we introduce another thing, let's call it C, a thing that is different from both A and B, then C wouldn't be X for it isn't "Either A or B". And noone would be able to say, without expressing a falsehood, that C is an X.

Be wary of terms such as "completely different". What does that mean? If you have two sets A and B, where A = { 2, 4, 6 } and B = { 8, 10, 12 }, are they completely different? If and only if by "completely different" you mean "have no elements in common". But if you mean literally, as in, they have absolutely nothing in common, that's blatantly false. Both sets consist of exclusively even numbers. They are identical in this regard.

When we "identify two different things" and "say they are one thing", we are not necessarily lying, we're merely saying they share a sameness of some sort.

Re: Mlodinow: We Invent the Laws of Nature

Posted: Tue Sep 19, 2023 10:15 pm
by Skepdick
Magnus Anderson wrote: Tue Sep 19, 2023 10:10 pm
Iwannaplato wrote:We could identify two completely different things and say they are one thing. I am not sure this makes them one thing.
Suppose we have two different things A and B. We can come up with a symbol X and assign it the concept captured by the sentence "Either A or B". That would make both A and B an X. That would make them the same in this particular regard -- they would both be X. In other words, they would both be "Either X or Y". And that has nothing to do with what anyone believes and what anyone desires. You're not free to decide that A isn't X or that B isn't X.

If we introduce another thing, let's call it C, a thing that is different from both A and B, then C wouldn't be X for it isn't "Either A or B". And noone would be able to say, without expressing a falsehood, that C is an X.

Be wary of terms such as "completely different". What does that mean? If you have two sets A and B, where A = { 2, 4, 6 } and B = { 8, 10, 12 }, are they completely different? If and only if by "completely different" you mean "have no elements in common". But if you mean literally, as in, they have absolutely nothing in common, that's blatantly false. Both sets consist of exclusively even numbers. They are the same in this regard.

When we "identify two different things" and "say they are one thing", we are not necessarily lying, we're merely saying they share a sameness of some sort.
The most formal, precise and rigorous treatment of identity is the univalence axiom.

https://ncatlab.org/nlab/show/univalence+axiom

(A = B) ≃ (A ≃ B). identity is equivalent to equivalence.

This single axiom can act as an alternative (to set theory) foundation to all of Mathematics.

https://en.wikipedia.org/wiki/Univalent_foundations

Re: Mlodinow: We Invent the Laws of Nature

Posted: Tue Sep 19, 2023 10:25 pm
by Skepdick
Magnus Anderson wrote: Tue Sep 19, 2023 10:10 pm When we "identify two different things" and "say they are one thing", we are not necessarily lying, we're merely saying they share a sameness of some sort.
Yes, your intuition is spot on. There is a connection between them.

https://en.wikipedia.org/wiki/Homotopy_ ... y#Equality